Simplifying Fractions: A Step-by-Step Guide

Understanding the Problem: Simplifying Fractions

Hey everyone! Let's dive into the world of fractions and learn how to solve and reduce them to their simplest form. This is super important for math, and it's something you'll use a lot, trust me! Our goal is to figure out the answer to this problem: $\left(\frac{9}{14}+\frac{6}{14}\right) \times\left(\frac{3}{5} \div \frac{1}{9}\right)$. The most important thing to remember is to provide the final answer as a fraction, not a decimal. So, no calculators allowed here, unless you're using them to check your work! Let's start with the basics. Fractions are a way to represent parts of a whole. The top number (numerator) tells you how many parts you have, and the bottom number (denominator) tells you how many parts make up the whole. For example, in the fraction $\frac{1}{2}$, you have 1 part out of a total of 2 parts. When we solve fractional expressions like this one, we're essentially combining these parts and performing mathematical operations on them. It might seem complicated at first, but I promise, with a bit of practice, it will become second nature. The key is to break the problem down into smaller, manageable steps. We need to follow the order of operations (PEMDAS/BODMAS), which means we'll handle parentheses first, then multiplication and division (from left to right), and finally, addition and subtraction (from left to right). Remember to be patient with yourself, and don't be afraid to ask questions. Mathematics is a journey, and the more you practice, the more comfortable you'll become. Understanding fractions is fundamental to various mathematical concepts. From calculating areas and volumes to understanding proportions and ratios, a solid grasp of fractions will serve as a building block for your mathematical journey. Mastering fractions not only equips you with the skills to tackle complex mathematical problems but also enhances your problem-solving abilities in everyday situations. Whether you are dividing a pizza among friends, adjusting a recipe, or understanding financial concepts, the ability to work with fractions is invaluable. Let's get started and make fractions your friend! So, let's break this down and conquer this problem step by step, shall we? This is going to be fun, so let's enjoy the process and celebrate our progress along the way.

Step 1: Solving the Addition Inside the Parentheses

Alright, let's start with the first part of the problem: $\left(\frac{9}{14}+\frac{6}{14}\right)$. See, we have two fractions, $\frac{9}{14}$ and $\frac{6}{14}$, and we need to add them together. The good news is that they already have the same denominator (14). This makes our job much easier, you guys! When adding fractions with the same denominator, you simply add the numerators and keep the denominator the same. So, we have 9 + 6 = 15. Therefore, $\frac{9}{14}+\frac{6}{14} = \frac{15}{14}$. That was pretty straightforward, right? Now, we can simplify the first part of our expression. Remember, fractions represent division, and we can perform basic arithmetic operations to combine them. For instance, if we have $\frac{1}{2}$ and $\frac{1}{2}$, we can add them to get $\frac{2}{2}$, which is equal to 1. In our case, both fractions have the same denominator. This makes adding them a piece of cake! You just need to add the numerators and keep the denominator constant. If the denominators are different, we would have to find the least common multiple (LCM) of the denominators and convert the fractions so they have a common denominator before we can add or subtract them. But don't worry, we'll get to that later if we need to. This is a crucial step because it simplifies our equation, making it easier to handle subsequent operations. Always remember, the goal is to reduce the complexity step by step. By addressing the addition within the parentheses first, we are setting ourselves up for the next stage which is division, which will make the final calculation easier. Now that we've successfully solved the addition, we can move on to the next part of the expression, which involves division. Ready for the next step?

Step 2: Solving the Division Inside the Second Parentheses

Okay, let's move on to the second part of our original equation: $\left(\frac{3}{5} \div \frac{1}{9}\right)$. Here, we have a division problem, and it's between two fractions. Remember, when you divide fractions, you actually multiply by the reciprocal of the second fraction. The reciprocal is just flipping the fraction upside down. So, the reciprocal of $\frac{1}{9}$ is $\frac{9}{1}$. Therefore, $\frac{3}{5} \div \frac{1}{9}$ becomes $\frac{3}{5} \times \frac{9}{1}$. Now, to multiply fractions, you multiply the numerators and multiply the denominators. So, 3 times 9 equals 27, and 5 times 1 equals 5. That means $\frac{3}{5} \times \frac{9}{1} = \frac{27}{5}$. Easy, huh? Let's recap. Dividing fractions is all about flipping and multiplying. First, we change the division to multiplication by inverting the second fraction (the divisor). Then we multiply the numerators (top numbers) to get the new numerator of the product, and multiply the denominators (bottom numbers) to get the new denominator of the product. That's it! This step is essential because it allows us to simplify the expression, eventually leading to a single fraction that represents the final result. The key to solving fraction division problems is to be precise and accurate when finding the reciprocal and performing the multiplication. If you make a mistake in either of these steps, your answer will be incorrect. Always double-check your calculations to avoid common errors. Now that we've conquered the division, we're ready to put all the pieces together! We have successfully handled both the addition and division parts of our original equation. The next step involves bringing the two results together through multiplication.

Step 3: Multiplying the Results

We're almost there, guys! Now that we've solved the addition and the division, we have $\frac{15}{14}$ and $\frac{27}{5}$. Remember, the original problem was $\left(\frac{9}{14}+\frac{6}{14}\right) \times\left(\frac{3}{5} \div \frac{1}{9}\right)$. Now we know that $\left(\frac{9}{14}+\frac{6}{14}\right) = \frac{15}{14}$ and $\left(\frac{3}{5} \div \frac{1}{9}\right) = \frac{27}{5}$. So the next step is to multiply these two fractions together: $\frac{15}{14} \times \frac{27}{5}$. To multiply fractions, we multiply the numerators and the denominators. So, 15 times 27 equals 405, and 14 times 5 equals 70. That gives us $\frac{405}{70}$. We're almost done! This step is the culmination of our efforts. Now we combine everything and move toward the final solution. The key here is to ensure accurate multiplication of the fractions. One common mistake is multiplying the wrong numbers or forgetting a step in the process. Double-checking your calculations and maintaining precision will help you avoid these pitfalls. We’re on the final stretch, and we're about to reach the finish line. Before we declare victory, we need to simplify our result and give it our final touch. The last step is to reduce the fraction to its lowest terms.

Step 4: Reducing the Fraction to Lowest Terms

Okay, we have our answer: $\frac{405}{70}$. But wait! We always need to reduce the fraction to its lowest terms. This means finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by that number. In other words, find the largest number that can divide both 405 and 70 without any remainders. The GCD of 405 and 70 is 5. So, we divide both the numerator and the denominator by 5. 405 divided by 5 is 81, and 70 divided by 5 is 14. Therefore, $\frac{405}{70}$ simplifies to $\frac{81}{14}$. And that, my friends, is our final answer! Remember, reducing fractions to their lowest terms is super important. It makes the answer cleaner and easier to understand. It's like simplifying a sentence; we want to make it as clear and concise as possible. Always look for common factors to divide the numerator and the denominator. Always ensure that the fraction cannot be simplified further by dividing both by a common factor. This might involve checking if both the numerator and denominator are even or divisible by 3, 5, or 7. In this case, after dividing by 5, the numbers 81 and 14 have no common factors other than 1, so the fraction is fully reduced. Now that you have all of the steps laid out, remember that practice makes perfect. So, keep practicing, and don't hesitate to ask for help. The journey of mathematics is exciting, so embrace the challenges, and enjoy the process of learning and growing! Congratulations! You have successfully solved the problem and reduced the fraction to its simplest form. This skill will be helpful in all of your mathematical endeavors! Well done!